we give a criterion for the hypoellipticity of a class of differentialoperators of hormander type$$l = 1/2sum_{i = 0}^n x_i^2 + x_0.$$our criterion is weaker than the usual finite-type hormander condition and allows for decencies of $l$ of exponential order on hypersurfaces in the ambient space. the methods of proof are probabilistic, being based upon an analysis of the regularity of the diffusion process driven by $l$. these methods are also used to study the dirichlet problem associated with exponentially degenerate operators $l$.